section \<open>Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space.\<close>

 theory Cartesian_Euclidean_Space

 imports Finite_Cartesian_Product Integration

 begin

 lemma subspace_special_hyperplane: "subspace {x. x $ k = 0}"

   by (simp add: subspace_def)

 lemma delta_mult_idempotent:

   "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)"

   by simp

 lemma setsum_UNIV_sum:

   fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"

   shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"

   apply (subst UNIV_Plus_UNIV [symmetric])

   apply (subst setsum.Plus)

   apply simp_all

   done

 lemma setsum_mult_product:

   "setsum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"

   unfolding setsum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric]

 proof (rule setsum.cong, simp, rule setsum.reindex_cong)

   fix i

   show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)

   show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"

   proof safe

     fix j assume "j \<in> {i * B..<i * B + B}"

     then show "j \<in> (\<lambda>j. j + i * B) ` {..<B}"

       by (auto intro!: image_eqI[of _ _ "j - i * B"])

   qed simp

 qed simp

 subsection\<open>Basic componentwise operations on vectors.\<close>

 instantiation vec :: (times, finite) times

 begin

 definition "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"

 instance ..

 end

 instantiation vec :: (one, finite) one

 begin

 definition "1 \<equiv> (\<chi> i. 1)"

 instance ..

 end

 instantiation vec :: (ord, finite) ord

 begin

 definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"

 definition "x < (y::'a^'b) \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"

 instance ..

 end

 text\<open>The ordering on one-dimensional vectors is linear.\<close>

 class cart_one =

   assumes UNIV_one: "card (UNIV :: 'a set) = Suc 0"

 begin

 subclass finite

 proof

   from UNIV_one show "finite (UNIV :: 'a set)"

     by (auto intro!: card_ge_0_finite)

 qed

 end

 instance vec:: (order, finite) order

   by standard (auto simp: less_eq_vec_def less_vec_def vec_eq_iff

       intro: order.trans order.antisym order.strict_implies_order)

 instance vec :: (linorder, cart_one) linorder

 proof

   obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"

   proof -

     have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one)

     then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)

     then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto

     then show thesis by (auto intro: that)

   qed

   fix x y :: "'a^'b::cart_one"

   note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps

   show "x \<le> y \<or> y \<le> x" by auto

 qed

 text\<open>Constant Vectors\<close>

 definition "vec x = (\<chi> i. x)"

 lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b"

   by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis)

 text\<open>Also the scalar-vector multiplication.\<close>

 definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)

   where "c *s x = (\<chi> i. c * (x$i))"

 subsection \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space.\<close>

 lemma setsum_cong_aux:

   "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"

   by (auto intro: setsum.cong)

 hide_fact (open) setsum_cong_aux

 method_setup vector = \<open>

 let

   val ss1 =

     simpset_of (put_simpset HOL_basic_ss @{context}

       addsimps [@{thm setsum.distrib} RS sym,

       @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},

       @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym])

   val ss2 =

     simpset_of (@{context} addsimps

              [@{thm plus_vec_def}, @{thm times_vec_def},

               @{thm minus_vec_def}, @{thm uminus_vec_def},

               @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},

               @{thm scaleR_vec_def},

               @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}])

   fun vector_arith_tac ctxt ths =

     simp_tac (put_simpset ss1 ctxt)

     THEN' (fn i => resolve_tac ctxt @{thms Cartesian_Euclidean_Space.setsum_cong_aux} i

          ORELSE resolve_tac ctxt @{thms setsum.neutral} i

          ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i)

     (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)

     THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths)

 in

   Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (vector_arith_tac ctxt ths))

 end

 \<close> "lift trivial vector statements to real arith statements"

 lemma vec_0[simp]: "vec 0 = 0" by vector

 lemma vec_1[simp]: "vec 1 = 1" by vector

 lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector

 lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto

 lemma vec_add: "vec(x + y) = vec x + vec y"  by vector

 lemma vec_sub: "vec(x - y) = vec x - vec y" by vector

 lemma vec_cmul: "vec(c * x) = c *s vec x " by vector

 lemma vec_neg: "vec(- x) = - vec x " by vector

 lemma vec_setsum:

   assumes "finite S"

   shows "vec(setsum f S) = setsum (vec \<circ> f) S"

   using assms

 proof induct

   case empty

   then show ?case by simp

 next

   case insert

   then show ?case by (auto simp add: vec_add)

 qed

 text\<open>Obvious "component-pushing".\<close>

 lemma vec_component [simp]: "vec x $ i = x"

   by vector

 lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"

   by vector

 lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"

   by vector

 lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector

 lemmas vector_component =

   vec_component vector_add_component vector_mult_component

   vector_smult_component vector_minus_component vector_uminus_component

   vector_scaleR_component cond_component

 subsection \<open>Some frequently useful arithmetic lemmas over vectors.\<close>

 instance vec :: (semigroup_mult, finite) semigroup_mult

   by standard (vector mult.assoc)

 instance vec :: (monoid_mult, finite) monoid_mult

   by standard vector+

 instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult

   by standard (vector mult.commute)

 instance vec :: (comm_monoid_mult, finite) comm_monoid_mult

   by standard vector

 instance vec :: (semiring, finite) semiring

   by standard (vector field_simps)+

 instance vec :: (semiring_0, finite) semiring_0

   by standard (vector field_simps)+

 instance vec :: (semiring_1, finite) semiring_1

   by standard vector

 instance vec :: (comm_semiring, finite) comm_semiring

   by standard (vector field_simps)+

 instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..

 instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..

 instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..

 instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..

 instance vec :: (ring, finite) ring ..

 instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..

 instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..

 instance vec :: (ring_1, finite) ring_1 ..

 instance vec :: (real_algebra, finite) real_algebra

   by standard (simp_all add: vec_eq_iff)

 instance vec :: (real_algebra_1, finite) real_algebra_1 ..

 lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"

 proof (induct n)

   case 0

   then show ?case by vector

 next

   case Suc

   then show ?case by vector

 qed

 lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) $ i = 1"

   by vector

 lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) $ i = - 1"

   by vector

 instance vec :: (semiring_char_0, finite) semiring_char_0

 proof

   fix m n :: nat

   show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"

     by (auto intro!: injI simp add: vec_eq_iff of_nat_index)

 qed

 instance vec :: (numeral, finite) numeral ..

 instance vec :: (semiring_numeral, finite) semiring_numeral ..

 lemma numeral_index [simp]: "numeral w $ i = numeral w"

   by (induct w) (simp_all only: numeral.simps vector_add_component one_index)

 lemma neg_numeral_index [simp]: "- numeral w $ i = - numeral w"

   by (simp only: vector_uminus_component numeral_index)

 instance vec :: (comm_ring_1, finite) comm_ring_1 ..

 instance vec :: (ring_char_0, finite) ring_char_0 ..

 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"

   by (vector mult.assoc)

 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"

   by (vector field_simps)

 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"

   by (vector field_simps)

 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector

 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector

 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"

   by (vector field_simps)

 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector

 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector

 lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector

 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector

 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"

   by (vector field_simps)

 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"

   by (simp add: vec_eq_iff)

 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)

 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"

   by vector

 lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"

   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)

 lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"

   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)

 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"

   by (metis vector_mul_lcancel)

 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"

   by (metis vector_mul_rcancel)

 lemma component_le_norm_cart: "\<bar>x$i\<bar> <= norm x"

   apply (simp add: norm_vec_def)

   apply (rule member_le_setL2, simp_all)

   done

 lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x$i\<bar> <= e"

   by (metis component_le_norm_cart order_trans)

 lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"

   by (metis component_le_norm_cart le_less_trans)

 lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"

   by (simp add: norm_vec_def setL2_le_setsum)

 lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"

   unfolding scaleR_vec_def vector_scalar_mult_def by simp

 lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"

   unfolding dist_norm scalar_mult_eq_scaleR

   unfolding scaleR_right_diff_distrib[symmetric] by simp

 lemma setsum_component [simp]:

   fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"

   shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"

 proof (cases "finite S")

   case True

   then show ?thesis by induct simp_all

 next

   case False

   then show ?thesis by simp

 qed

 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"

   by (simp add: vec_eq_iff)

 lemma setsum_cmul:

   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"

   shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"

   by (simp add: vec_eq_iff setsum_right_distrib)

 lemma setsum_norm_allsubsets_bound_cart:

   fixes f:: "'a \<Rightarrow> real ^'n"

   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"

   shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"

   using setsum_norm_allsubsets_bound[OF assms]

   by simp

 subsection\<open>Closures and interiors of halfspaces\<close>

 lemma interior_halfspace_le [simp]:

   assumes "a \<noteq> 0"

     shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"

 proof -

   have *: "a \<bullet> x < b" if x: "x \<in> S" and S: "S \<subseteq> {x. a \<bullet> x \<le> b}" and "open S" for S x

   proof -

     obtain e where "e>0" and e: "cball x e \<subseteq> S"

       using \<open>open S\<close> open_contains_cball x by blast

     then have "x + (e / norm a) *\<^sub>R a \<in> cball x e"

       by (simp add: dist_norm)

     then have "x + (e / norm a) *\<^sub>R a \<in> S"

       using e by blast

     then have "x + (e / norm a) *\<^sub>R a \<in> {x. a \<bullet> x \<le> b}"

       using S by blast

     moreover have "e * (a \<bullet> a) / norm a > 0"

       by (simp add: \<open>0 < e\<close> assms)

     ultimately show ?thesis

       by (simp add: algebra_simps)

   qed

   show ?thesis

     by (rule interior_unique) (auto simp: open_halfspace_lt *)

 qed

 lemma interior_halfspace_ge [simp]:

    "a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}"

 using interior_halfspace_le [of "-a" "-b"] by simp

 lemma interior_halfspace_component_le [simp]:

      "interior {x. x$k \<le> a} = {x :: (real,'n::finite) vec. x$k < a}" (is "?LE")

   and interior_halfspace_component_ge [simp]:

      "interior {x. x$k \<ge> a} = {x :: (real,'n::finite) vec. x$k > a}" (is "?GE")

 proof -

   have "axis k (1::real) \<noteq> 0"

     by (simp add: axis_def vec_eq_iff)

   moreover have "axis k (1::real) \<bullet> x = x$k" for x

     by (simp add: cart_eq_inner_axis inner_commute)

   ultimately show ?LE ?GE

     using interior_halfspace_le [of "axis k (1::real)" a]

           interior_halfspace_ge [of "axis k (1::real)" a] by auto

 qed

 lemma closure_halfspace_lt [simp]:

   assumes "a \<noteq> 0"

     shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"

 proof -

   have [simp]: "-{x. a \<bullet> x < b} = {x. a \<bullet> x \<ge> b}"

     by (force simp:)

   then show ?thesis

     using interior_halfspace_ge [of a b] assms

     by (force simp: closure_interior)

 qed

 lemma closure_halfspace_gt [simp]:

    "a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}"

 using closure_halfspace_lt [of "-a" "-b"] by simp

 lemma closure_halfspace_component_lt [simp]:

      "closure {x. x$k < a} = {x :: (real,'n::finite) vec. x$k \<le> a}" (is "?LE")

   and closure_halfspace_component_gt [simp]:

      "closure {x. x$k > a} = {x :: (real,'n::finite) vec. x$k \<ge> a}" (is "?GE")

 proof -

   have "axis k (1::real) \<noteq> 0"

     by (simp add: axis_def vec_eq_iff)

   moreover have "axis k (1::real) \<bullet> x = x$k" for x

     by (simp add: cart_eq_inner_axis inner_commute)

   ultimately show ?LE ?GE

     using closure_halfspace_lt [of "axis k (1::real)" a]

           closure_halfspace_gt [of "axis k (1::real)" a] by auto

 qed

 lemma interior_hyperplane [simp]:

   assumes "a \<noteq> 0"

     shows "interior {x. a \<bullet> x = b} = {}"

 proof -

   have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"

     by (force simp:)

   then show ?thesis

     by (auto simp: assms)

 qed

 lemma frontier_halfspace_le:

   assumes "a \<noteq> 0 \<or> b \<noteq> 0"

     shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"

 proof (cases "a = 0")

   case True with assms show ?thesis by simp

 next

   case False then show ?thesis

     by (force simp: frontier_def closed_halfspace_le)

 qed

 lemma frontier_halfspace_ge:

   assumes "a \<noteq> 0 \<or> b \<noteq> 0"

     shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}"

 proof (cases "a = 0")

   case True with assms show ?thesis by simp

 next

   case False then show ?thesis

     by (force simp: frontier_def closed_halfspace_ge)

 qed

 lemma frontier_halfspace_lt:

   assumes "a \<noteq> 0 \<or> b \<noteq> 0"

     shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}"

 proof (cases "a = 0")

   case True with assms show ?thesis by simp

 next

   case False then show ?thesis

     by (force simp: frontier_def interior_open open_halfspace_lt)

 qed

 lemma frontier_halfspace_gt:

   assumes "a \<noteq> 0 \<or> b \<noteq> 0"

     shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}"

 proof (cases "a = 0")

   case True with assms show ?thesis by simp

 next

   case False then show ?thesis

     by (force simp: frontier_def interior_open open_halfspace_gt)

 qed

 lemma interior_standard_hyperplane:

    "interior {x :: (real,'n::finite) vec. x$k = a} = {}"

 proof -

   have "axis k (1::real) \<noteq> 0"

     by (simp add: axis_def vec_eq_iff)

   moreover have "axis k (1::real) \<bullet> x = x$k" for x

     by (simp add: cart_eq_inner_axis inner_commute)

   ultimately show ?thesis

     using interior_hyperplane [of "axis k (1::real)" a]

     by force

 qed

 subsection \<open>Matrix operations\<close>

 text\<open>Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\<close>

 definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"

     (infixl "**" 70)

   where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"

 definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"

     (infixl "*v" 70)

   where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"

 definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "

     (infixl "v*" 70)

   where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"

 definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"

 definition transpose where

   "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"

 definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"

 definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"

 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"

 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"

 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)

 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"

   by (vector matrix_matrix_mult_def setsum.distrib[symmetric] field_simps)

 lemma matrix_mul_lid:

   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"

   shows "mat 1 ** A = A"

   apply (simp add: matrix_matrix_mult_def mat_def)

   apply vector

   apply (auto simp only: if_distrib cond_application_beta setsum.delta'[OF finite]

     mult_1_left mult_zero_left if_True UNIV_I)

   done

 lemma matrix_mul_rid:

   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"

   shows "A ** mat 1 = A"

   apply (simp add: matrix_matrix_mult_def mat_def)

   apply vector

   apply (auto simp only: if_distrib cond_application_beta setsum.delta[OF finite]

     mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)

   done

 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"

   apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult.assoc)

   apply (subst setsum.commute)

   apply simp

   done

 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"

   apply (vector matrix_matrix_mult_def matrix_vector_mult_def

     setsum_right_distrib setsum_left_distrib mult.assoc)

   apply (subst setsum.commute)

   apply simp

   done

 lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"

   apply (vector matrix_vector_mult_def mat_def)

   apply (simp add: if_distrib cond_application_beta setsum.delta' cong del: if_weak_cong)

   done

 lemma matrix_transpose_mul:

     "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"

   by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)

 lemma matrix_eq:

   fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"

   shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")

   apply auto

   apply (subst vec_eq_iff)

   apply clarify

   apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)

   apply (erule_tac x="axis ia 1" in allE)

   apply (erule_tac x="i" in allE)

   apply (auto simp add: if_distrib cond_application_beta axis_def

     setsum.delta[OF finite] cong del: if_weak_cong)

   done

 lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"

   by (simp add: matrix_vector_mult_def inner_vec_def)

 lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"

   apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib ac_simps)

   apply (subst setsum.commute)

   apply simp

   done

 lemma transpose_mat: "transpose (mat n) = mat n"

   by (vector transpose_def mat_def)

 lemma transpose_transpose: "transpose(transpose A) = A"

   by (vector transpose_def)

 lemma row_transpose:

   fixes A:: "'a::semiring_1^_^_"

   shows "row i (transpose A) = column i A"

   by (simp add: row_def column_def transpose_def vec_eq_iff)

 lemma column_transpose:

   fixes A:: "'a::semiring_1^_^_"

   shows "column i (transpose A) = row i A"

   by (simp add: row_def column_def transpose_def vec_eq_iff)

 lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"

   by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)

 lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"

   by (metis transpose_transpose rows_transpose)

 text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>

 lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"

   by (simp add: matrix_vector_mult_def inner_vec_def)

 lemma matrix_mult_vsum:

   "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"

   by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)

 lemma vector_componentwise:

   "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"

   by (simp add: axis_def if_distrib setsum.If_cases vec_eq_iff)

 lemma basis_expansion: "setsum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"

   by (auto simp add: axis_def vec_eq_iff if_distrib setsum.If_cases cong del: if_weak_cong)

 lemma linear_componentwise:

   fixes f:: "real ^'m \<Rightarrow> real ^ _"

   assumes lf: "linear f"

   shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")

 proof -

   let ?M = "(UNIV :: 'm set)"

   let ?N = "(UNIV :: 'n set)"

   have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"

     unfolding setsum_component by simp

   then show ?thesis

     unfolding linear_setsum_mul[OF lf, symmetric]

     unfolding scalar_mult_eq_scaleR[symmetric]

     unfolding basis_expansion

     by simp

 qed

 text\<open>Inverse matrices  (not necessarily square)\<close>

 definition

   "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"

 definition

   "matrix_inv(A:: 'a::semiring_1^'n^'m) =

     (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"

 text\<open>Correspondence between matrices and linear operators.\<close>

 definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"

   where "matrix f = (\<chi> i j. (f(axis j 1))$i)"

 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"

   by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff

       field_simps setsum_right_distrib setsum.distrib)

 lemma matrix_works:

   assumes lf: "linear f"

   shows "matrix f *v x = f (x::real ^ 'n)"

   apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)

   apply clarify

   apply (rule linear_componentwise[OF lf, symmetric])

   done

 lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"

   by (simp add: ext matrix_works)

 lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"

   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)

 lemma matrix_compose:

   assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"

     and lg: "linear (g::real^'m \<Rightarrow> real^_)"

   shows "matrix (g \<circ> f) = matrix g ** matrix f"

   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]

   by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)

 lemma matrix_vector_column:

   "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"

   by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)

 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"

   apply (rule adjoint_unique)

   apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def

     setsum_left_distrib setsum_right_distrib)

   apply (subst setsum.commute)

   apply (auto simp add: ac_simps)

   done

 lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"

   shows "matrix(adjoint f) = transpose(matrix f)"

   apply (subst matrix_vector_mul[OF lf])

   unfolding adjoint_matrix matrix_of_matrix_vector_mul

   apply rule

   done

 subsection \<open>lambda skolemization on cartesian products\<close>

 (* FIXME: rename do choice_cart *)

 lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>

    (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")

 proof -

   let ?S = "(UNIV :: 'n set)"

   { assume H: "?rhs"

     then have ?lhs by auto }

   moreover

   { assume H: "?lhs"

     then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis

     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"

     { fix i

       from f have "P i (f i)" by metis

       then have "P i (?x $ i)" by auto

     }

     hence "\<forall>i. P i (?x$i)" by metis

     hence ?rhs by metis }

   ultimately show ?thesis by metis

 qed

 lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"

   unfolding inner_simps scalar_mult_eq_scaleR by auto

 lemma left_invertible_transpose:

   "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"

   by (metis matrix_transpose_mul transpose_mat transpose_transpose)

 lemma right_invertible_transpose:

   "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"

   by (metis matrix_transpose_mul transpose_mat transpose_transpose)

 lemma matrix_left_invertible_injective:

   "(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"

 proof -

   { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"

     from xy have "B*v (A *v x) = B *v (A*v y)" by simp

     hence "x = y"

       unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }

   moreover

   { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"

     hence i: "inj (op *v A)" unfolding inj_on_def by auto

     from linear_injective_left_inverse[OF matrix_vector_mul_linear i]

     obtain g where g: "linear g" "g \<circ> op *v A = id" by blast

     have "matrix g ** A = mat 1"

       unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]

       using g(2) by (simp add: fun_eq_iff)

     then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }

   ultimately show ?thesis by blast

 qed

 lemma matrix_left_invertible_ker:

   "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"

   unfolding matrix_left_invertible_injective

   using linear_injective_0[OF matrix_vector_mul_linear, of A]

   by (simp add: inj_on_def)

 lemma matrix_right_invertible_surjective:

   "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"

 proof -

   { fix B :: "real ^'m^'n"

     assume AB: "A ** B = mat 1"

     { fix x :: "real ^ 'm"

       have "A *v (B *v x) = x"

         by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }

     hence "surj (op *v A)" unfolding surj_def by metis }

   moreover

   { assume sf: "surj (op *v A)"

     from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]

     obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A \<circ> g = id"

       by blast

     have "A ** (matrix g) = mat 1"

       unfolding matrix_eq  matrix_vector_mul_lid

         matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]

       using g(2) unfolding o_def fun_eq_iff id_def

       .

     hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast

   }

   ultimately show ?thesis unfolding surj_def by blast

 qed

 lemma matrix_left_invertible_independent_columns:

   fixes A :: "real^'n^'m"

   shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>

       (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"

     (is "?lhs \<longleftrightarrow> ?rhs")

 proof -

   let ?U = "UNIV :: 'n set"

   { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"

     { fix c i

       assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"

       let ?x = "\<chi> i. c i"

       have th0:"A *v ?x = 0"

         using c

         unfolding matrix_mult_vsum vec_eq_iff

         by auto

       from k[rule_format, OF th0] i

       have "c i = 0" by (vector vec_eq_iff)}

     hence ?rhs by blast }

   moreover

   { assume H: ?rhs

     { fix x assume x: "A *v x = 0"

       let ?c = "\<lambda>i. ((x$i ):: real)"

       from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]

       have "x = 0" by vector }

   }

   ultimately show ?thesis unfolding matrix_left_invertible_ker by blast

 qed

 lemma matrix_right_invertible_independent_rows:

   fixes A :: "real^'n^'m"

   shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>

     (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"

   unfolding left_invertible_transpose[symmetric]

     matrix_left_invertible_independent_columns

   by (simp add: column_transpose)

 lemma matrix_right_invertible_span_columns:

   "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>

     span (columns A) = UNIV" (is "?lhs = ?rhs")

 proof -

   let ?U = "UNIV :: 'm set"

   have fU: "finite ?U" by simp

   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"

     unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def

     apply (subst eq_commute)

     apply rule

     done

   have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast

   { assume h: ?lhs

     { fix x:: "real ^'n"

       from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"

         where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast

       have "x \<in> span (columns A)"

         unfolding y[symmetric]

         apply (rule span_setsum)

         unfolding scalar_mult_eq_scaleR

         apply (rule span_mul)

         apply (rule span_superset)

         unfolding columns_def

         apply blast

         done

     }

     then have ?rhs unfolding rhseq by blast }

   moreover

   { assume h:?rhs

     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"

     { fix y

       have "?P y"

       proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])

         show "\<exists>x::real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"

           by (rule exI[where x=0], simp)

       next

         fix c y1 y2

         assume y1: "y1 \<in> columns A" and y2: "?P y2"

         from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"

           unfolding columns_def by blast

         from y2 obtain x:: "real ^'m" where

           x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast

         let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"

         show "?P (c*s y1 + y2)"

         proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong)

           fix j

           have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)

               else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"

             using i(1) by (simp add: field_simps)

           have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)

               else (x$xa) * ((column xa A$j))) ?U = setsum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"

             apply (rule setsum.cong[OF refl])

             using th apply blast

             done

           also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"

             by (simp add: setsum.distrib)

           also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"

             unfolding setsum.delta[OF fU]

             using i(1) by simp

           finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)

             else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .

         qed

       next

         show "y \<in> span (columns A)"

           unfolding h by blast

       qed

     }

     then have ?lhs unfolding lhseq ..

   }

   ultimately show ?thesis by blast

 qed

 lemma matrix_left_invertible_span_rows:

   "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"

   unfolding right_invertible_transpose[symmetric]

   unfolding columns_transpose[symmetric]

   unfolding matrix_right_invertible_span_columns

   ..

 text \<open>The same result in terms of square matrices.\<close>

 lemma matrix_left_right_inverse:

   fixes A A' :: "real ^'n^'n"

   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"

 proof -

   { fix A A' :: "real ^'n^'n"

     assume AA': "A ** A' = mat 1"

     have sA: "surj (op *v A)"

       unfolding surj_def

       apply clarify

       apply (rule_tac x="(A' *v y)" in exI)

       apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)

       done

     from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]

     obtain f' :: "real ^'n \<Rightarrow> real ^'n"

       where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast

     have th: "matrix f' ** A = mat 1"

       by (simp add: matrix_eq matrix_works[OF f'(1)]

           matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])

     hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp

     hence "matrix f' = A'"

       by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)

     hence "matrix f' ** A = A' ** A" by simp

     hence "A' ** A = mat 1" by (simp add: th)

   }

   then show ?thesis by blast

 qed

 text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>

 definition "rowvector v = (\<chi> i j. (v$j))"

 definition "columnvector v = (\<chi> i j. (v$i))"

 lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"

   by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)

 lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"

   by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)

 lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"

   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)

 lemma dot_matrix_product:

   "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"

   by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)

 lemma dot_matrix_vector_mul:

   fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"

   shows "(A *v x) \<bullet> (B *v y) =

       (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"

   unfolding dot_matrix_product transpose_columnvector[symmetric]

     dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..

 lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |i. i\<in>UNIV}"

   by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)

 lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"

   using Basis_le_infnorm[of "axis i 1" x]

   by (simp add: Basis_vec_def axis_eq_axis inner_axis)

 lemma continuous_component: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"

   unfolding continuous_def by (rule tendsto_vec_nth)

 lemma continuous_on_component: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"

   unfolding continuous_on_def by (fast intro: tendsto_vec_nth)

 lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"

   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)

 lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"

   unfolding bounded_def

   apply clarify

   apply (rule_tac x="x $ i" in exI)

   apply (rule_tac x="e" in exI)

   apply clarify

   apply (rule order_trans [OF dist_vec_nth_le], simp)

   done

 lemma compact_lemma_cart:

   fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"

   assumes f: "bounded (range f)"

   shows "\<exists>l r. subseq r \<and>

         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"

     (is "?th d")

 proof -

   have "\<forall>d' \<subseteq> d. ?th d'"

     by (rule compact_lemma_general[where unproj=vec_lambda])

       (auto intro!: f bounded_component_cart simp: vec_lambda_eta)

   then show "?th d" by simp

 qed

 instance vec :: (heine_borel, finite) heine_borel

 proof

   fix f :: "nat \<Rightarrow> 'a ^ 'b"

   assume f: "bounded (range f)"

   then obtain l r where r: "subseq r"

       and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"

     using compact_lemma_cart [OF f] by blast

   let ?d = "UNIV::'b set"

   { fix e::real assume "e>0"

     hence "0 < e / (real_of_nat (card ?d))"

       using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto

     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"

       by simp

     moreover

     { fix n

       assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"

       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"

         unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)

       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"

         by (rule setsum_strict_mono) (simp_all add: n)

       finally have "dist (f (r n)) l < e" by simp

     }

     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

       by (rule eventually_mono)

   }

   hence "((f \<circ> r) \<longlongrightarrow> l) sequentially" unfolding o_def tendsto_iff by simp

   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto

 qed

 lemma interval_cart:

   fixes a :: "real^'n"

   shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"

     and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"

   by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)

 lemma mem_interval_cart:

   fixes a :: "real^'n"

   shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"

     and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"

   using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)

 lemma interval_eq_empty_cart:

   fixes a :: "real^'n"

   shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)

     and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)

 proof -

   { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>box a b"

     hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval_cart by auto

     hence "a$i < b$i" by auto

     hence False using as by auto }

   moreover

   { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"

     let ?x = "(1/2) *\<^sub>R (a + b)"

     { fix i

       have "a$i < b$i" using as[THEN spec[where x=i]] by auto

       hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"

         unfolding vector_smult_component and vector_add_component

         by auto }

     hence "box a b \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }

   ultimately show ?th1 by blast

   { fix i x assume as:"b$i < a$i" and x:"x\<in>cbox a b"

     hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval_cart by auto

     hence "a$i \<le> b$i" by auto

     hence False using as by auto }

   moreover

   { assume as:"\<forall>i. \<not> (b$i < a$i)"

     let ?x = "(1/2) *\<^sub>R (a + b)"

     { fix i

       have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto

       hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"

         unfolding vector_smult_component and vector_add_component

         by auto }

     hence "cbox a b \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }

   ultimately show ?th2 by blast

 qed

 lemma interval_ne_empty_cart:

   fixes a :: "real^'n"

   shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"

     and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"

   unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)

     (* BH: Why doesn't just "auto" work here? *)

 lemma subset_interval_imp_cart:

   fixes a :: "real^'n"

   shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"

     and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b"

     and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> cbox a b"

     and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b"

   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart

   by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)

 lemma interval_sing:

   fixes a :: "'a::linorder^'n"

   shows "{a .. a} = {a} \<and> {a<..<a} = {}"

   apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)

   done

 lemma subset_interval_cart:

   fixes a :: "real^'n"

   shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)

     and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)

     and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)

     and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)

   using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)

 lemma disjoint_interval_cart:

   fixes a::"real^'n"

   shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)

     and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)

     and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)

     and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)

   using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)

 lemma inter_interval_cart:

   fixes a :: "real^'n"

   shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"

   unfolding inter_interval

   by (auto simp: mem_box less_eq_vec_def)

     (auto simp: Basis_vec_def inner_axis)

 lemma closed_interval_left_cart:

   fixes b :: "real^'n"

   shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"

   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)

 lemma closed_interval_right_cart:

   fixes a::"real^'n"

   shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"

   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)

 lemma is_interval_cart:

   "is_interval (s::(real^'n) set) \<longleftrightarrow>

     (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"

   by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)

 lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"

   by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)

 lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"

   by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)

 lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"

   by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)

 lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"

   by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)

 lemma Lim_component_le_cart:

   fixes f :: "'a \<Rightarrow> real^'n"

   assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x $i \<le> b) net"

   shows "l$i \<le> b"

   by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])

 lemma Lim_component_ge_cart:

   fixes f :: "'a \<Rightarrow> real^'n"

   assumes "(f \<longlongrightarrow> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"

   shows "b \<le> l$i"

   by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])

 lemma Lim_component_eq_cart:

   fixes f :: "'a \<Rightarrow> real^'n"

   assumes net: "(f \<longlongrightarrow> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"

   shows "l$i = b"

   using ev[unfolded order_eq_iff eventually_conj_iff] and

     Lim_component_ge_cart[OF net, of b i] and

     Lim_component_le_cart[OF net, of i b] by auto

 lemma connected_ivt_component_cart:

   fixes x :: "real^'n"

   shows "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s.  z$k = a)"

   using connected_ivt_hyperplane[of s x y "axis k 1" a]

   by (auto simp add: inner_axis inner_commute)

 lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"

   unfolding subspace_def by auto

 lemma closed_substandard_cart:

   "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"

 proof -

   { fix i::'n

     have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"

       by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) }

   thus ?thesis

     unfolding Collect_all_eq by (simp add: closed_INT)

 qed

 lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"

   (is "dim ?A = _")

 proof -

   let ?a = "\<lambda>x. axis x 1 :: real^'n"

   have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"

     by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)

   have "?a ` d \<subseteq> Basis"

     by (auto simp: Basis_vec_def)

   thus ?thesis

     using dim_substandard[of "?a ` d"] card_image[of ?a d]

     by (auto simp: axis_eq_axis inj_on_def *)

 qed

 lemma affinity_inverses:

   assumes m0: "m \<noteq> (0::'a::field)"

   shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"

   "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"

   using m0

   apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)

   apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])

   done

 lemma vector_affinity_eq:

   assumes m0: "(m::'a::field) \<noteq> 0"

   shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"

 proof

   assume h: "m *s x + c = y"

   hence "m *s x = y - c" by (simp add: field_simps)

   hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp

   then show "x = inverse m *s y + - (inverse m *s c)"

     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)

 next

   assume h: "x = inverse m *s y + - (inverse m *s c)"

   show "m *s x + c = y" unfolding h

     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)

 qed

 lemma vector_eq_affinity:

     "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"

   using vector_affinity_eq[where m=m and x=x and y=y and c=c]

   by metis

 lemma vector_cart:

   fixes f :: "real^'n \<Rightarrow> real"

   shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"

   unfolding euclidean_eq_iff[where 'a="real^'n"]

   by simp (simp add: Basis_vec_def inner_axis)

 lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"

   by (rule vector_cart)

 subsection "Convex Euclidean Space"

 lemma Cart_1:"(1::real^'n) = \<Sum>Basis"

   using const_vector_cart[of 1] by (simp add: one_vec_def)

 declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]

 declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]

 lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component

 lemma convex_box_cart:

   assumes "\<And>i. convex {x. P i x}"

   shows "convex {x. \<forall>i. P i (x$i)}"

   using assms unfolding convex_def by auto

 lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"

   by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval)

 lemma unit_interval_convex_hull_cart:

   "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}"

   unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]

   by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)

 lemma cube_convex_hull_cart:

   assumes "0 < d"

   obtains s::"(real^'n) set"

     where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"

 proof -

   from assms obtain s where "finite s"

     and "cbox (x - setsum (op *\<^sub>R d) Basis) (x + setsum (op *\<^sub>R d) Basis) = convex hull s"

     by (rule cube_convex_hull)

   with that[of s] show thesis

     by (simp add: const_vector_cart)

 qed

 subsection "Derivative"

 definition "jacobian f net = matrix(frechet_derivative f net)"

 lemma jacobian_works:

   "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>

     (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"

   apply rule

   unfolding jacobian_def

   apply (simp only: matrix_works[OF linear_frechet_derivative]) defer

   apply (rule differentiableI)

   apply assumption

   unfolding frechet_derivative_works

   apply assumption

   done

 subsection \<open>Component of the differential must be zero if it exists at a local

   maximum or minimum for that corresponding component.\<close>

 lemma differential_zero_maxmin_cart:

   fixes f::"real^'a \<Rightarrow> real^'b"

   assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"

     "f differentiable (at x)"

   shows "jacobian f (at x) $ k = 0"

   using differential_zero_maxmin_component[of "axis k 1" e x f] assms

     vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]

   by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)

 subsection \<open>Lemmas for working on @{typ "real^1"}\<close>

 lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"

   by (metis (full_types) num1_eq_iff)

 lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"

   by auto (metis (full_types) num1_eq_iff)

 lemma exhaust_2:

   fixes x :: 2

   shows "x = 1 \<or> x = 2"

 proof (induct x)

   case (of_int z)

   then have "0 <= z" and "z < 2" by simp_all

   then have "z = 0 | z = 1" by arith

   then show ?case by auto

 qed

 lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"

   by (metis exhaust_2)

 lemma exhaust_3:

   fixes x :: 3

   shows "x = 1 \<or> x = 2 \<or> x = 3"

 proof (induct x)

   case (of_int z)

   then have "0 <= z" and "z < 3" by simp_all

   then have "z = 0 \<or> z = 1 \<or> z = 2" by arith

   then show ?case by auto

 qed

 lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"

   by (metis exhaust_3)

 lemma UNIV_1 [simp]: "UNIV = {1::1}"

   by (auto simp add: num1_eq_iff)

 lemma UNIV_2: "UNIV = {1::2, 2::2}"

   using exhaust_2 by auto

 lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"

   using exhaust_3 by auto

 lemma setsum_1: "setsum f (UNIV::1 set) = f 1"

   unfolding UNIV_1 by simp

 lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"

   unfolding UNIV_2 by simp

 lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"

   unfolding UNIV_3 by (simp add: ac_simps)

 instantiation num1 :: cart_one

 begin

 instance

 proof

   show "CARD(1) = Suc 0" by auto

 qed

 end

 subsection\<open>The collapse of the general concepts to dimension one.\<close>

 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"

   by (simp add: vec_eq_iff)

 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"

   apply auto

   apply (erule_tac x= "x$1" in allE)

   apply (simp only: vector_one[symmetric])

   done

 lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"

   by (simp add: norm_vec_def)

 lemma norm_real: "norm(x::real ^ 1) = \<bar>x$1\<bar>"

   by (simp add: norm_vector_1)

 lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x$1) - (y$1)\<bar>"

   by (auto simp add: norm_real dist_norm)

 subsection\<open>Explicit vector construction from lists.\<close>

 definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"

 lemma vector_1: "(vector[x]) $1 = x"

   unfolding vector_def by simp

 lemma vector_2:

  "(vector[x,y]) $1 = x"

  "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"

   unfolding vector_def by simp_all

 lemma vector_3:

  "(vector [x,y,z] ::('a::zero)^3)$1 = x"

  "(vector [x,y,z] ::('a::zero)^3)$2 = y"

  "(vector [x,y,z] ::('a::zero)^3)$3 = z"

   unfolding vector_def by simp_all

 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"

   apply auto

   apply (erule_tac x="v$1" in allE)

   apply (subgoal_tac "vector [v$1] = v")

   apply simp

   apply (vector vector_def)

   apply simp

   done

 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"

   apply auto

   apply (erule_tac x="v$1" in allE)

   apply (erule_tac x="v$2" in allE)

   apply (subgoal_tac "vector [v$1, v$2] = v")

   apply simp

   apply (vector vector_def)

   apply (simp add: forall_2)

   done

 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"

   apply auto

   apply (erule_tac x="v$1" in allE)

   apply (erule_tac x="v$2" in allE)

   apply (erule_tac x="v$3" in allE)

   apply (subgoal_tac "vector [v$1, v$2, v$3] = v")

   apply simp

   apply (vector vector_def)

   apply (simp add: forall_3)

   done

 lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"

   apply (rule bounded_linearI[where K=1])

   using component_le_norm_cart[of _ k] unfolding real_norm_def by auto

 lemma integral_component_eq_cart[simp]:

   fixes f :: "'n::euclidean_space \<Rightarrow> real^'m"

   assumes "f integrable_on s"

   shows "integral s (\<lambda>x. f x $ k) = integral s f $ k"

   using integral_linear[OF assms(1) bounded_linear_component_cart,unfolded o_def] .

 lemma interval_split_cart:

   "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"

   "cbox a b \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"

   apply (rule_tac[!] set_eqI)

   unfolding Int_iff mem_interval_cart mem_Collect_eq interval_cbox_cart

   unfolding vec_lambda_beta

   by auto

 end